VORTICAL FLOWS IN AN ADVERSE PRESSURE GRADIENT by John M. Brookfield Submitted to the Department of Aeronautics and Astronautics in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 1993 © Massachusetts Institute of Technology 1993 All rights reserved Signature of Author Departrnt of Aeronautics and Astronautics May 17,1993 Certified by __ Profess~'r Edward M. Greitzer Thesis Supervisor, Director of Gas Turbine Laboratory Professor of Aeronautics and Astronautics Certified by Prqse dIan A. Waitz Professor of Aeronautics and Astronautics A Certified by fessor Harold Y. Wachman Chairman, partment Graduate Committee MASSACHUSETTS INSTITUTE SOF TFr#4PJJ nnhy 'jUN 08 1993 UBRARIES VORTICAL FLOWS IN AN ADVERSE PRESSURE GRADIENT by John M. Brookfield Submitted to the Department of Aeronautics and Astronautics on May 17, 1993 in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics and Astronautics Abstract An analytical and computational study was performed to examine the effects of adverse pressure gradients on confined vortical motions representative of those encountered in compressor clearance flows. The study included exploring the parametric dependence of vortex core size and blockage, as well as designing an experimental configuration to examine this phenomena in a controlled manner. Two approaches were used to assess the flows of interest. A one-dimensional analysis was developed to evaluate the parametric dependence of vortex core expansion in a diffusing duct as a function of core to diffuser radius ratio, axial velocity ratio (core to outer flow) and swirl ratio (tangential to axial velocity at the core edge) at the inlet. It was found that a critical swirl ratio exists, above which core expansion and total pressure losses were large. Tip clearance flow in a curved, diffusing duct designed to simulate a turbomachine blade passage was examined using a three-dimensional Navier-Stokes code. Both overall behavior, such as pressure rise, and more detailed aspects (e.g. clearance vortex trajectory) were investigated. Over the range of parameters evaluated, the blockage of the clearance flow increased linearly with pressure gradient. The combined results of the one- dimensional analysis and the computational simulation suggest that the abrupt expansion of the vortex core which is associated with the phenomenon of vortex breakdown is unlikely in turbomachine compressors. Thesis Supervisor: Dr. Edward M. Greitzer Title: Professor of Aeronautics and Astronautics Director of Gas Turbine Laboratory ACKNOWLEDGEMENTS I wish to thank Prof. Edward Greitzer for his constant support and enthusiasm throughout the research. His insight into vortical flows was inspirational to understanding the resulting tip clearance flows. In addition, I would like to thank the entire tip clearance group in the Gas Turbine Laboratory, which additionally consisted of Prof. Ian Waitz, Dr. Choon Tan, Dr. Gavin Hendricks, Arif Khalid, Andy Rothstein, and others. The lively group discussions were crucial to understanding many aspects of the vortical flows. Each of these people also deserve individual thanks for contributions to the author's research, with particular recognition going to Arif Khalid and the use of his programs to examine the computational flow fields. I would also like to thank Prof. Marten Landahl, Prof. Nick Cumpsty, and Prof. Frank Marble for their additional insight and understanding, and Dr. John Adamczyk and Mark Celestina at NASA Lewis Research Center for their help in modifying the numerical code. Only the combined guidance of all the people above made this research possible. Lastly, I would like to thank my family for their continual support, and G. Scott McNulty for many interesting and enlightening discussions. This research was funded by NASA Lewis Research Center under grant NAG3-1237, Mr. Gary J. Skoch, Technical Monitor, and by the Air Force Research in Aeropropulsion Technology Program. This support is gratefully acknowledged TABLE OF CONTENTS page Chapter 1 INTRODUCTION 1.1 Background to the Problem 1.2 Organization Chapter 2 ONE-DIMENSIONAL ANALYSIS 2.1 Background 2.2 Basic Flow Model 2.3 Solution Procedure 2.4 Qualitative Discussion of Predicted Trends 2.5 Parametric Study 2.5.1 Variation of Velocity Ratio 2.5.2 Core Pressure Rise 2.5.3 Critical Swirl Ratio 2.6 Core Mixing Losses 2.6.1 Mixing of Isentropically Expanded Vortex Cores 2.6.2 Estimate of Mixing Loss for Large Swirl 2.7 Effects of Core Dissipation 2.8 Comparison with Equivalent Unidirectional Flow 2.8.1 Developing an Equivalent Unidirectional Flow 2.8.2 Isentropic Expansion 2.8.3 Scaling for Equal Core Mass Flow 33 2.8.4 Mixing of the Isentropically Expanded Flows 34 2.9 Summary and Conclusions for the One-Dimensional Analysis 35 Chapter 3 CONCEPTUAL DESIGN OF A TIP CLEARANCE FLOW EXPERIMENT: Background, Solution Method, and Evaluation Tools 3.1 Introduction 46 3.2 Simplified Clearance Flow Configuration 47 3.3 Computational Flow Code 48 3.4 Code Modifications 48 3.4.1 Boundary Conditions 48 3.4.2 Symmetry Plane 49 3.4.3 Turbulence Model 49 3.5 Computational Grid 50 3.6 Design of Experimental Configuration 51 3.7 Vortex Core Trajectory 54 3.7.1 Similarity Solution 54 3.7.2 Importance of Clearance Flow Angle 55 3.8 Circulation of Clearance Flow Vortex 56 3.9 Blockage in Three-Dimensional Flows 57 Chapter 4 NUMERICAL RESULTS 4.1 Introduction 69 4.2 Straight Duct Geometry 69 4.2.1 Zero Turbulent Viscosity 69 4.2.2 Decrease in Circulation 71 4.2.3 Effects of Viscosity 73 4.3 Blade Passage Pressure Gradient 75 4.4 Relative Wall Motion Effects 76 4.5 Comparison of Core Trajectory with Trailing Edge Flow Field 77 4.6 Passage with Increased Pressure Gradient 78 4.7 Dependence of Pressure Rise on Clearance Flow Blockage 78 Chapter 5 Summary, Conclusions, and Recommendations for Future Work 5.1 Summary and Conclusions 95 5.2 Recommendations for Future Work 97 References Appendix A Flow Code Modifications A.1 Slot Flow Inlet Boundary Conditions 99 A.2 Inviscid Boundary Conditions for a Slanted Surface 101 Appendix B Computational Grid Study 104 Appendix C Schwartz-Christoffel Transformation 110 LIST OF FIGURES page Chapter 2 2.1 Schematic of one-dimensional analysis 36 2.2 Radius and velocity ratios (local/inlet) for axial velocity ratio (u/U) = 2 and swirl ratio (V/U) = 1.4 at the inlet 37 2.3 Radius and velocity ratios (local/inlet) for axial velocity ratio (u/U) = .5 and swirl ratio (V/U) = .1 at the inlet 37 2.4 Core radius ratios (exit/inlet) for three different inlet axial velocity ratios as a function of inlet swirl ratio 38 2.5 Core axial velocity ratios (exit/inlet) for three different inlet axial velocity ratios as a function of inlet swirl ratio 38 2.6 Growth of core swirl ratio (based on core axial velocity) through the diffusing duct for an inlet axial velocity ratio (u/U) of one and two different inlet swirl ratios (V/U) of 0.4 and 0.7 39 2.7 Pressure rise ratio (core centerline to core edge) for three inlet axial velocity ratios and varying inlet swirl ratio 39 2.8 Vortex core expansion showing critical radius ratio of 2: u/U = 1 and V/U = .7 at the inlet 40 2.9 Schematic of step change in circular duct (four to one area ratio) 40 2.10 Net loss of total pressure versus number of steps used to mix flow for a step change in circular duct (four to one area ratio) 41 2.11 Loss in total pressure due to mixing of isentropically expanded Rankine vortex for cases with small core expansion (one step) 41 2.12 Schematic of unidirectional flow model for comparing mixing losses of small and large core expansion 42 2.13 Effects of core dissipation on vortex core expansion with inlet axial velocity ratio (u/U) of one 42 2.14 Comparison of swirling and equivalent unidirectional velocity and pressure profiles 43 2.15 Isentropically expanded swirling and corresponding unidirectional flow radius ratios (exit/inlet) for three different inlet axial velocity ratios and varying inlet swirl 44 2.16 Scaled isentropically expanded flows (equal core mass flows) 44 2.17 Final exit mixed radius ratios of swirling and unidirectional flow for three different inlet axial velocity ratios and varying inlet swirl 45 2.18 Total pressure loss due to mixing (one step) of isentropically expanded swirling and unidirectional flow 45 Chapter 3 3.1 Model of tip clearance flow using slender body assumptions, (Chen, 1991). Comparison of a) three-dimensional steady flow, and b) two-dimensional unsteady flow model 59 3.2 Tip clearance vortex modeling philosophy with transverse injection and inviscid symmetry plane: a) overall configuration, b) slot injection, and c) tip clearance flow 60 3.3 Schematic of cell centered scheme inviscid wall, dummy cell boundary condition 60 3.4 Schematic of calculation strategy for wake turbulent model. Each line parallel to the blade surface is calculated separately with the symmetry plane used as the wake center 61 3.5 Methodology for creating quasi-rectangular grid cross section using cylindrical coordinates required for computational code 61 3.6 Computational grid used for straight duct geometry. Grid stretching on blade surfaces, in slot region, and at leading and trailing edges: a) radial plane, b) tangential plane, c) axial plane 62 3.7 Computed static pressure contours on a blade passage radial plane, (Khalid, 1993) 63 3.8 Blade surface static pressure distribution at mid-span with no tip clearance (Storer, 1991) 63 3.9 Computed static pressure contours